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Forecasting seasonal time series data: a Bayesian model averaging approach

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Abstract

A flexible Bayesian periodic autoregressive model is used for the prediction of quarterly and monthly time series data. As the unknown autoregressive lag order, the occurrence of structural breaks and their respective break dates are common sources of uncertainty these are treated as random quantities within the Bayesian framework. Since no analytical expressions for the corresponding marginal posterior predictive distributions exist a Markov Chain Monte Carlo approach based on data augmentation is proposed. Its performance is demonstrated in Monte Carlo experiments. Instead of resorting to a model selection approach by choosing a particular candidate model for prediction, a forecasting approach based on Bayesian model averaging is used in order to account for model uncertainty and to improve forecasting accuracy. For model diagnosis a Bayesian sign test is introduced to compare the predictive accuracy of different forecasting models in terms of statistical significance. In an empirical application, using monthly unemployment rates of Germany, the performance of the model averaging prediction approach is compared to those of model selected Bayesian and classical (non)periodic time series models.

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Notes

  1. In the empirical analysis below \(c_{1}\) and \(c_{2}\) in (4) are both set equal to 100 in order to express lack of prior knowledge with regard to the variation of the regression coefficients.

  2. In the following the first p observations are used as initial values \(\mathbf {y}_{0}\). The conditioning on \(\mathbf {y}_{0}\) is suppressed subsequently.

  3. In case of the discrete break date \(T_{B}\) the corresponding integration is in fact a summation.

  4. For \(m=0\) this step is omitted.

  5. For a PAR(p) model a stationarity condition can be stated by using a multivariate model representation as in Tiao and Grupe (1980).

  6. Here the variable of interest is simply regressed on a set of S dummy variables \(D_{s,t}\), which equal one if observation t is associated with season s.

  7. Note that the MAPE for a specific horizon k does not depend on the scale or dimension.

  8. Note that the sign test presumes i.i.d. observations, an assumption that needs to be checked in practice.

  9. This is the \(S_{2}\)-test statistic of Diebold and Mariano (1995), p. 255, which follows a Binomial distribution with parameters T and \(\pi _{i,j}=0.5\) under the null hypothesis.

  10. For example, in the MC experiments presented below, 2-years ahead forecasts using quarterly data are conducted and thus \(T=8\), whereas in the empirical application of Sect. 5, 1-year ahead forecasts using monthly data are considered and thus the length of the realized loss-differential sequences is \(T=12\).

  11. Here for all computations \(\alpha =\beta =10^{-10}\) is used.

  12. Here \(\omega _{0}=0.5\) is chosen.

  13. The SARMA specification corresponds to the ‘constant parameter representation’ of a monthly PAR(1) process, cf. Ghysels and Osborn (2001), p. 150 for details.

  14. All initial values are chosen to be fixed and equal to zero.

  15. The MC integration steps to obtain the marginal posterior predictive distributions of the \(y_{T+k},~k=1 \ldots 8,\) are conducted on a grid of 100 points.

  16. For each loss differential series a Runs test for randomness is conducted, where rejection of the null of ‘randomness’ would be problematic with regard to the iid-assumption of the used tests, see Diebold and Mariano (1995). Here no further evidence for nonrandomness of the sequences has been found.

  17. Similar results have been obtained for other parameterizations of the DGP in (18) and also for a periodic moving average process of order one as DGP. The average PMSEs in the latter case are 1.22 for a PAR(1) model, 1.23 for an AR(1) model and 1.26 for a PMEANS model.

  18. The corresponding results under Haldane’s prior are 0.0077 for all three comparisons.

  19. The SARMA model has an averaged PMSE (MAPE) of 1.87 (2.84).

  20. This reform brought together the former unemployment benefits for long term unemployed (‘Arbeitslosenhilfe’) and the former welfare benefits (‘Sozialhilfe’). That is, since January 2005 these two groups have both been considered as ‘unemployed’. This simple change in ‘measurement’ of the unemployment rate induced the instantaneous level shift for most of the series.

  21. In this context, note the following useful approximate relationship between the BIC and the posterior probability mass function of model \(M_i\): \(f(M_{i}| ~data) \approx \exp {(-1/2~ BIC_{i})}/\sum _{j=1}^{I} \exp {(-1/2~ BIC_{j})}\), which can be derived by applying a Laplace approximation (see Tierney and Kadane 1986; Tierney et al. 1989) to the joint posterior density.

  22. Most of these six loss-differences are however not statistically significant.

  23. The results for the 18 series are omitted here.

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Authors and Affiliations

  1. Chair for Statistics and Econometrics, Institute for Employment Research, Nuremberg, University of Erlangen-Nuremberg, Erlangen, Germany

    Alexander Vosseler

  2. Institute for Employment Research (IAB), Nuremberg, University of Regensburg, Regensburg, Germany

    Enzo Weber

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  1. Alexander Vosseler
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  2. Enzo Weber
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Correspondence to Enzo Weber.

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Appendices

Appendix A

See Tables 1, 2, 3, 4, 5, 6, 7 and 8.

Table 1 Test results—design 1/DGP: quarterly PAR(1)
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Table 2 Test results—design 2/DGP: PIAR(1)
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Table 3 Test results—design 3/DGP: SAR(1)
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Table 4 Test results—design 4/DGP: SARMA\((1,0)\times (1,1)\)
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Table 5 Test results—design 5/DGP: monthly PAR(1)
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Table 6 Testing for no periodicity
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Table 7 Evaluation of 12-months ahead forecasts
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Table 8 Evaluation of 12-months ahead forecasts (Cont.)
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Appendix B

See Figs. 1, 2, 3, 4, 5, 6, 7, 8 and 9.

Fig. 1
figure 1

(Cumulated) PMSEs for 8-quarters ahead forecasts (design 1 and 2)

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Fig. 2
figure 2

(Cumulated) PMSEs for 8-quarters ahead forecasts (design 3 and 4)

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Fig. 3
figure 3

(Cumulated) PMSEs for 12-months ahead forecasts (design 5)

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Fig. 4
figure 4

Posterior probability of \(H_{1}\) as a function of x for \(T=8\)

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Fig. 5
figure 5

Posterior probability of \(H_{1}\) as a function of x for \(T=60\)

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Fig. 6
figure 6

One-year ahead forecasts of the unemployment rates of West-Germany

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Fig. 7
figure 7

Model averaged posterior predictive densities of West-Germany (1)

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Fig. 8
figure 8

Model averaged posterior predictive densities of West-Germany (2)

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Fig. 9
figure 9

Model averaged posterior predictive densities of West-Germany (3)

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Vosseler, A., Weber, E. Forecasting seasonal time series data: a Bayesian model averaging approach. Comput Stat 33, 1733–1765 (2018). https://doi.org/10.1007/s00180-018-0801-3

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